Topics in Room Acoustics

1. Topics in Room Acoustics

2. Outline • Review of absorption coefficient and absorptivity. • Derivation of reverb time formula. • Standing wave resonance in 1-, 2-, and 3-dimensions. Room modes. • Modifying an acoustic space: the physics of • slat absorbers • diffusers.

3. Room Acoustics Intro • Much of basic acoustics is a simplified model that assumes that free field conditions exist. • In free field the SPL or SIL drops off 6 dB every time distance from the source is doubled. (Review example). • The presence of an enclosure alters free field conditions: • Multiple reflections lead to reverberation (>200 Hz) • Closed path reflections lead to standing wave resonances— Room Modes (<200 Hz)

4. Room parameters • Dimensions—height, width, length and shape of the room (these values imply room volume). • How the surfaces reflect sound is determined by the wall material and its preparation. This quantity is described by the absorption coefficient, a. • The properties of the whole room are described by the sum of absorption coefficients weighted by their areal contribution to the room--the absorptivity, A.

5. Absorptivity • Absorptivity formula • Example: Room 3 m tall with floor and ceiling 8 m x 5 m. aceil=0.3, aflr=0.6, awalls=0.12. • What is A? • What is weighted average a? [with a, A = a x total surface area] • Remember a depends on frequency.

6. Statistical model of reverb time • Statistical model assumes that the entire room is uniformly filled with sound energy. The sound has repeated collisions with the walls losing energy with each collision as determined by a. • In a room with volume V and interior surface area S the average number of collisions per second, n, is given by

7. Derivation of room energy after time, t • E(t), the energy left in room after a time, t, (i.e. after nt collisions) is

8. Reverb time definition • Reverb time, Tr, is defined as the time for the sound energy to drop by 106. Thus, • Solving for Tr • In metric units

9. Waetzmann-Schuster-Eyring reverb time formula If a is small then this formula approximates to the more familiar Sabine form (from Physics 1600) ln(1-a)~a

10. When does the statistical model apply? • Statistical model applies to “large” rooms ones in which the reverberant field dominates the properties of the room. • A reverberant or diffuse field is one in which the time-averaged sound pressure is equal everywhere in the room. Sound energy flow is equally probable in all directions. • In a “small room” the resonant standing waves—the so-called room modes dominate the response.

11. Room modes • Room modes refer to the standing wave resonances that exist in an enclosed space. • To visualize the standing wave modes recall the resonant modes on a string. When a resonant frequency excites the string a standing wave is set up with nodes and antinodes. The resonant frequencies are harmonic. • In 2 and 3 dimensions similar standing waves exist but the resonant frequencies are not harmonically related.

12. Standing waves in a rectangular enclosure • Modes are described by mode numbers n1, n2, n3 • Room dimensions are L (length), W (width), and H (height).

13. Examples • Large room 12mx4mx8m Frequency of mode resonance

14. Example • Small room 4m x 5m x 3m Frequency of mode resonance

15. Semi-reverberant room calculations • A room that has a mix of reverberant sound and direct sound from a source is called semi-reverberant. • Note that most real rooms are semi-reverberant. • The sound in many parts of the room is reverberant with energy flow equal in all directions (far from the sound source); however, near the source, the sound flow is directional.

16. Sound source calculations • Non-directional sound source in free field. At distance R from source, direct sound is • Directional sound source (Q is directivity) • Where W is the watts of acoustic power from source and W0=1x10-12 Watts

17. Directivity factor • The directivity factor Q is a measure of the directional nature of a sound source. Q is defined as the ratio of intensity from the directional source, Id, divided by the intensity of an omnidirectional source, I0. • Directivity Index (DI) is Q expressed in dB.

18. Q due to wall and corner reflections

19. Reverberant sound • Far from the source the decibel level of the reverberant sound is given by • Example—noise reduction. Change A from 45 Sabins to 120 Sabins. What is the change in reverberant sound of a 10-3 Watt source.

20. Direct and reverberant sound • Combined formula for both direct and reverberant sound

21. Critical distance • The critical distance, Dc, is the distance at which the direct and reverberant sound levels are equal. • Equal when • Thus,

22. Why is critical distance important? • Speech intelligibility • For distances from the source much greater than the critical distance, speech becomes increasingly more difficult to understand because most of the sound energy comes from reflections. %ALCONS measures the loss of understanding of consonants. • Microphone placement • General rule: microphone should be no more that 0.3Dc for omnidirectional mic. 0.5Dc for directional mic.

23. %Articulation Loss of Consonants • %ALCONS formula • R Distance from speaker to listener • Tr Reverb time • Q directivity factor • V room volume • n number of reinforcing loudspeakers

24. %Articulation Loss of Consonants • What does the %ALCONS number mean? • Low numbers are good, that means very few (as a percentage) misunderstood consonants. • 10% is good • 15% is the limit beyond which intelligibility decreases • As we will show later (and Wheel of Fortune proves every night) language is redundant—we don’t need all the consonants to get meaning.

25. Large Room Example • Room dimensions 12 m x 14 m x 6 m • a = 0.2 • Calculate A and Tr. • What are the lowest 5 standing wave frequencies? • If a 3x10-2 W average output acoustic source is placed in the center of the front wall find • The reverberant level in dB • The total db at a distance of 3 m from the source • The critical distance • %ALCONS at R=3 m, 9 m, and at 15 m from the source.

26. Early Reflections • The timing of the first reflection is an important aesthetic parameter in auditorium acoustics. Why? No physical reason that I have seen! • We know (from MATLAB demos) that if the first reflection is delayed by greater than about 35 ms then we hear an echo—an undesirable effect. • Best values obtained by evaluating “good” concert halls are less than 35 ms. 20 ms for an “intimate” hall.

27. Precedence or Haas effect • Even in the presence of reflections we can localize the sound source. If similar sounds arrive at the ear within 35 ms the direction of the source is the direction of the first arriving sound. Note that we only hear one sound—not an echo which would need a longer delay of 65 ms or so. • Localization review—for frequencies up to 1kHz localization is due to inter-aural differences in phase (continuous signal) or in time delay (clicks). For >4kHz inter-aural intensity difference (diffraction around the head). In between some combination.

28. Small Room Acoustics • Early reflections are REALLY early because the walls and ceiling are so close. • Rooms may be reverberant in that 4/A > Q/4pR2, but the reverb time TR is short. Example in Homework. • Standing waves modes are well separated at low frequencies leading to very uneven low frequency response.

29. To see a page with a room calculator using many of the concepts we have developed go to • http://www.mcsquared.com/ssdesgnm.htm#calculate

30. Diffusers • Diffusers are used to minimize strong specular reflections in a small room. • Aim: eliminate specular reflection and replace it with diffuse scattering.

31. How do diffusers work? • Two basic methods • Random scattering from a roughened or textured surfaces. Easy to make but not predictable in response. • Diffraction by profiles that possess “all” necessary grating spacings to ensure a uniform diffraction pattern.

32. Maximal length sequence (MLS) • Binary profile—limited usefulness in practice • MLS sequences have other uses that we may explore in the MATLAB sessions. • Simple example: seed -1-1-1 with simple multiplication algorithm generates sequence -1-1-1+1+1-1+1-1-1-1 • This sequence contains all the grating combinations of 3-length gratings.

33. Quadratic residue method • A method of designing a multilevel diffuser that operates over a greater wavelength range. • Sequence of depths dn is generated by • Where the sequence sn is defined by

34. Well width and diffuser bandwidth • Maximum well depth should be 1.5 times wavelength of lowest frequency of operations • Well width should be 0.5 the wavelength of the highest frequency of operation • Highest frequency to lowest frequency define the operating bandwidth of the diffuser

35. Example • Choose design wavelength • Choose prime number seed, p • Generate sequence • Calculate depths • E.g 1000 Hz, p=13

36. History & applications of diffusers • Schroeder: maximal length sequences (1975); quadratic residue method (1979). • Small room applications –studios, including at MTSU. • Large auditoria—particularly for ceilings to suppress early ceiling reflection in favor of side wall reflection.

37. Damping low frequency standing wave modes in small rooms • A number of issues are considered with damping • Where to place damping material to get maximum loss—how does damping work. • Bass traps and slot absorber design—variations on the Helmholtz resonator.

38. Damping sound • Sound is damped by converting acoustic wave energy into heat usually by some form of friction. • Soft, porous materials are useful for damping high frequencies because air can move through BUT the moving air suffers multiple collisions with the foamy material. • Wood panels, dry wall etc move with low pressure waves and absorb low frequency energy. • Key feature—for high frequencies foam must be placed where displacement amplitude (same as particle velocity) is large.

39. Porous and Edge absorbers • Absorber effectiveness depends on the position of the materials with respect to the reflecting surface. • Max. velocity is at l/4. • Porous material close to a wall does not damp low frequencies—e.g. fabric curtains vs carpet.

40. Room Mode: Pressure profile

41. Room Mode: Displacement Profile

42. Slot Absorber • One example of absorber based on Helmholtz resonator • Slotted panel that is spaced away from one of the walls of the enclosure.

43. Helmholtz Resonator • Trapped air acts as a spring • Air in the neck acts as the mass. (vs is the speed of sound)

44. Slot absorber is a HR! • Fraction of open area, e • In one repeat distance V=AtotD, thus • Plug Aopen/V into HR formula

45. Slot absorber • Resonance frequency f is given by

46. Uses of the Slot Absorber • Reduce low frequency reverb time without affecting high frequency reverb time. • Suppress low frequency standing wave resonances—tunable! • Absorption can be varied by placement of foam either close to opening or set back between the wall and the slats.

47. Perforated panel absorber • Yet another version of the damped Helmholtz resonator (no foam damping!). • You can do the math to verify HR-ness! • p=perforation percentage; D=air space; t=effective hole length (panel thickness +0.8*hole diameter) Use meters for all measurements.

48. Industrial Panel absorber • Absorption coeff. • 125 Hz 0.22 • 250 Hz 0.77 • 500 Hz 1.12 • 1000 Hz 1.00 • 2000 Hz 0.78 • 4000 Hz 0.57

49. Panel absorber • Thin flexible plate (e.g. plywood) clamped at the edges. Low frequency pressure amplitude waves oscillate the plate—absorbing backing turns vibration to heat. • Plate has vibrational resonant frequencies. • Not an HR! • m mass per m2, D depth m

50. Final Thoughts • Room treatment depends greatly on the purpose of the space—classroom, musical auditorium, small vs large space… • Main parameters that affect experience—reverb time (large spaces), early reflections, standing wave resonances (small spaces). • Control methods—absorptivity, diffusers, low frequency traps